Sharp Circular Sampling and Derivative Period Polynomials

We determine the exact maximal reflected zero region that forces centered binomial samples of a balanced entire function to have all zeros on the unit circle. In degree $d\ge2$, this region is $$\Omega_d=\left\{a+ib:\ a^2-\frac{b^2}{d-1}\le\frac d4\right\}.$$ The finite theorem is sharp already for a single reflected zero pair, and a phase-preserving canonical-product approximation extends it to balanced entire functions of order at most one. De Bruijn strip contraction and projective Hermite--Kakeya--Obreschkoff theory then give the exact common-zero obstruction, simplicity, strict interlacing of consecutive derivative samples, and a monotone real-pencil root flow. As an application, we prove the derivative-period-polynomial unit-circle theorem for completed $L$-functions of primitive holomorphic newforms, in every derivative order and for arbitrary level and nebentypus. After the standard normalization, every zero of $$\sum_{j=0}^{k-2}\binom{k-2}{j}\Lambda^{(m)}(f,j+1)z^j$$ lies on the unit circle for every weight $k\ge4$, level, nebentypus, and derivative order $m\ge0$. In particular, this proves the full-polynomial unit-circle conjecture of Diamantis and Rolen in its original level-one setting and extends it to arbitrary level and nebentypus. The same source-side theorem also gives simplicity, strict interlacing, and, for each fixed derivative order, conductor-uniform quantitative localization in the weight aspect.